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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bertrand's theorem ： ウィキペディア英語版 Bertrand's theorem In classical mechanics, Bertrand's theorem states that among central force potentials with bound orbits, there are only two types of central force potentials with the property that all bound orbits are also closed orbits: (1) an inverse-square central force such as the gravitational or electrostatic potential :$V(\backslash mathbf)\; =\; \backslash frac,$ and (2) the radial harmonic oscillator potential :$V(\backslash mathbf)\; =\; \backslash frac\; kr^.$ The theorem was discovered by and named for Joseph Bertrand. ==General preliminaries== All attractive central forces can produce circular orbits, which are naturally closed orbits. The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circular orbits in general. The equation of motion for the radius ''r'' of a particle of mass ''m'' moving in a central potential ''V(r)'' is given by Lagrange's equations :$m\backslash frac\; =\; m\backslash frac\}$ where $\backslash omega\; \backslash equiv\; \backslash frac$ and the angular momentum ''L'' = ''mr''2ω is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force $\backslash frac$ equals the centripetal force requirement ''mr''ω2, as expected. The definition of angular momentum allows a change of independent variable from ''t'' to θ :$\backslash frac\; =\; \backslash frac$ giving the new equation of motion that is independent of time :$\backslash frac\; \backslash left(\; \backslash frac\; \backslash right)-\; \backslash frac\}\; =\; -\backslash frac$ This equation becomes quasilinear on making the change of variables $u\; \backslash equiv\; \backslash frac$ and multiplying both sides by $\backslash frac\}$ (see also Binet equation) :$\backslash frac\; V(1/u)$ As noted above, all central forces can produce circular orbits given an appropriate initial velocity. However, if some radial velocity is introduced, these orbits need not be stable (i.e., remain in orbit indefinitely) nor closed (repeatedly returning to exactly the same path). Here we show that stable, exactly closed orbits can be produced only with an inverse-square force or radial harmonic oscillator potential (a ''necessary condition''). In the following sections, we show that those force laws do produce stable, exactly closed orbits (a ''sufficient condition''). Define ''J(u)'' as :$\backslash frac\; V(1/u)\; =\; -\backslash frac\}\; f(1/u)$ where ''f'' represents the radial force. The criterion for perfectly circular motion at a radius ''r''0 is that the first term on the left-hand side be zero f(1/u_)|}} where $u\_\; \backslash equiv\; 1/r\_$. The next step is to consider the equation for ''u'' under ''small perturbations'' $\backslash eta\; \backslash equiv\; u\; -\; u\_$ from perfectly circular orbits. On the right-hand side, the ''J'' function can be expanded in a standard Taylor series :$J(u)\; \backslash approx\; J(u\_0)\; +\; \backslash eta\; J^(u\_)\; +\; \backslash frac\; \backslash eta^2\; J^(u\_0)\; +\; \backslash frac\; \backslash eta^3\; J^(u\_0)\; +\; \backslash cdots$ Substituting this expansion into the equation for ''u'' and subtracting the constant terms yields :$\backslash frac(u\_)\; +\; \backslash frac\; \backslash eta^\; J^(u\_)\; +\; \backslash frac\; \backslash eta^\; J^(u\_)\; +\; \backslash cdots$ which can be written as where $\backslash beta^\; \backslash equiv\; 1\; -\; J^(u\_)$ is a constant. β2 must be non-negative; otherwise, the radius of the orbit would vary exponentially away from its initial radius. (The solution β = 0 corresponds to a perfectly circular orbit.) If the right-hand side may be neglected (i.e., for small perturbations), the solutions are :$\backslash eta(\backslash theta)\; =\; h\_1\; \backslash cos\; \backslash left(\; \backslash beta\; \backslash theta\; \backslash right)$ where the amplitude ''h''1 is a constant of integration. For the orbits to be closed, β must be a rational number. What's more, it must be the ''same'' rational number for all radii, since β cannot change continuously; the rational numbers are totally disconnected from one another. Using the definition of ''J'' along with equation (1), : where $\backslash frac$ is evaluated at $(1/u\_)$. Since this must hold for any value of ''u''0, :$\backslash frac\; =\; \backslash left(\; \backslash beta^\; -\; 3\; \backslash right)\; \backslash frac$ which implies that the force must follow a power law :$f(r)\; =\; -\; \backslash frac$ Hence, ''J'' must have the general form u^} For more general deviations from circularity (i.e., when we cannot neglect the higher order terms in the Taylor expansion of ''J''), η may be expanded in a Fourier series, e.g., :$\backslash eta(\backslash theta)\; =\; h\_\; +\; h\_\; \backslash cos\; \backslash beta\; \backslash theta\; +\; h\_\; \backslash cos\; 2\backslash beta\; \backslash theta\; +\; h\_\; \backslash cos\; 3\backslash beta\; \backslash theta\; +\; \backslash cdots$ We substitute this into equation (2) and equate the coefficients belonging to the same frequency, keeping only the lowest order terms. As we show below, ''h''0 and ''h''2 are smaller than ''h''1, being of order $h\_1^2$. ''h''3, and all further coefficients, are at least of order $h\_1^3$. This makes sense since $h\_,\; h\_,\; h\_,\backslash ldots$ must all vanish faster than ''h''1 as a circular orbit is approached. :$h\_\; =\; h\_^\; \backslash frac)\}\; =\; -h\_^\; \backslash frac)\}\; =\; -\backslash frach\_\; \backslash frac)\}\; +\; h\_^\; \backslash frac)\}\; \backslash right\; )$ From the cos(βθ) term, we get :$0\; =\; \backslash left(\; 2\; h\_\; h\_\; +\; h\_\; h\_\; \backslash right)\; \backslash frac)\}\; +\; h\_^\; \backslash frac)\}\; =\; \backslash frac\; (3\; \backslash beta^2\; J^(u\_)\; +\; 5\; J^(u\_)^2)$ where in the last step we substituted in the values of ''h''0 and ''h''2. Using equations (3) and (1), we can calculate the second and third derivatives of ''J'' evaluated at ''u''0, :$J^(u\_)\; =\; -\backslash frac(u\_)\; =\; \backslash frac$ Substituting these values into the last equation yields the main result of Bertrand's theorem :$\backslash beta^\; \backslash left(\; 1\; -\; \backslash beta^\; \backslash right)\; \backslash left(\; 4\; -\; \backslash beta^\; \backslash right)\; =\; 0$ Hence, the only potentials that can produce stable, closed, non-circular orbits are the inverse-square force law (β = 1) and the radial harmonic oscillator potential (β = 2). The solution β = 0 corresponds to perfectly circular orbits, as noted above. ==Inverse-square force== For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written :$V(\backslash mathbf)\; =\; \backslash frac\; =\; -ku$ The orbit ''u''(θ) can be derived from the general equation :$\backslash frac\; V(1/u)\; =\; \backslash frac\; =\; \backslash frac\backslash right)\; \backslash right\; )$ where ''e'' (the eccentricity) and θ0 (the phase offset) are constants of integration. This is the general formula for a conic section that has one focus at the origin; ''e'' = 0 corresponds to a circle, ''e'' < 1 corresponds to an ellipse, ''e'' = 1 corresponds to a parabola, and ''e'' > 1 corresponds to a hyperbola. The eccentricity ''e'' is related to the total energy ''E'' (cf. the Laplace–Runge–Lenz vector) :$e\; =\; \backslash sqrtm\}\}$ Comparing these formulae shows that ''E'' < 0 corresponds to an ellipse, ''E'' = 0 corresponds to a parabola, and ''E'' > 0 corresponds to a hyperbola. In particular, $E=-\backslash frac\{2L^\{2\}\}$ for perfectly circular orbits. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bertrand's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.